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We prove the global $L^p$-boundedness of Fourier integral operators thatmodel the parametrices for hyperbolic partial differential equations, withamplitudes in classical H\"ormander classes $S^{m}_{\rho,\delta}(\mathbb{R}^n)$ for parameters $0<\rho\leq 1$, $0\leq \delta<1$. We alsoconsider the regularity of operators with amplitudes in the exotic class$S^{m}_{0, \delta}(\mathbb{R}^n)$, $0\leq \delta< 1$ and the forbidden class$S^{m}_{\rho, 1}(\mathbb{R}^n)$, $0\leq\rho\leq 1.$ Furthermore we show thatdespite the failure of the $L^2$-boundedness of operators with amplitudes inthe forbidden class $S^{0}_{1, 1}(\mathbb{R}^n)$, the operators in question arebounded on Sobolev spaces $H^s(\mathbb{R}^n)$ with $s>0.$ This result extendsthose of Y. Meyer and E. M. Stein to the setting of Fourier integral operators.

Regularity of Fourier integral operators with amplitudes in general H\"ormander classes